A227501 Number of non-congruent solutions of x^2 - xy + y^2 == 1 (mod n).

1, 3, 6, 6, 6, 18, 6, 12, 18, 18, 12, 36, 12, 18, 36, 24, 18, 54, 18, 36, 36, 36, 24, 72, 30, 36, 54, 36, 30, 108, 30, 48, 72, 54, 36, 108, 36, 54, 72, 72, 42, 108, 42, 72, 108, 72, 48, 144, 42, 90, 108, 72, 54, 162, 72, 72, 108, 90, 60, 216, 60, 90, 108, 96

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Multiplicative: If p == 2 (mod 3) then a(p^s) = (p+1)*p^(s-1); if p == 1 (mod 3) then a(p^s) = (p-1)*p^(s-1); a(3^s) = 2*3^s.

From Amiram Eldar, Oct 13 2022: (Start)

a(n) = 2 * A227128(n) if n divisible by 3, and a(n) = A227128(n) otherwise.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 2/(3 * A086724) = 0.853276... . (End)

Mathematica

Eisenstein[1] = 1; Eisenstein[n_] := Length@Select[Flatten[Table[{a, b}, {a, n}, {b, n}], 1], Mod[#[[1]]^2 + #[[2]]^2 - #[[1]]*#[[2]], n] == 1 &]; Array[Eisenstein, 100]
f[p_, e_] := If[Mod[p, 3] == 2, p + 1, p - 1]*p^(e - 1); f[3, e_] := 2*3^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 13 2022 *)

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 3, 2*3^f[i, 2], f[i, 1]^(f[i, 2] - 1) * (f[i, 1] + (-1)^(f[i, 1]%3))))}; \\ Amiram Eldar, Oct 13 2022

Crossrefs

Cf. A086724, A227128.

Sequence in context: A081289 A160504 A291801 * A294981 A079093 A153035

Adjacent sequences: A227498 A227499 A227500 * A227502 A227503 A227504

Keyword

nonn,mult

Author

José María Grau Ribas, Jul 13 2013

Status

approved